Eisenlohr’s optimal conformal map of the world

[Milo]

An interrupted Sinusoidal, in my opinion, is the best equal-area substitute for the Mercator, as it minimizes shape distortion. Things are sheared away from the central meridians, as one might expect, but the projection is accurate on the equator, and accurate along each central meridian.

Well, many people dislike having too many interruptions, even if they're all in the oceans. It does make it harder to tell how far apart the continents are supposed to be. But if that's not an issue, then yeah, interrupted pseudo-cylindrical maps work pretty well.

In a poor country, where each child can't be provided with his own atlas, the Mercator projection has the advantage that several children can go to the map at once, and see reasonable maps of different areas of the world - because it is conformal.

That's true of all conformal projections, though, so unless you really need your map to have straight-line meridians/parallels (either one implies the other, for conformal maps), you're better off with a projection that doesn't need infinite space to project the finite world. (I favor the Lagrange projection.)

This is the same reason why it was a sensible choice for Google Maps.

For modern computers, recomputing the map based on what part you're interested in takes no time at all (it doesn't even need to use actual projection math - adjusting a cylindrical equidistant or equal-area projection to have a different standard parallel can be done with nothing more than linear image scaling), so using a "one-size-fits-all" map is completely pointless whenever you have access to any computation resources more advanced than displaying a static image.

[Atarimaster]

I wonder why he made the 1st and 2nd projections useless/novelties when from his 3rd projection onwards we have actually useful world maps...

Regrettably, don’t have the original paper in which the six Eckert projections were introduced. Zöppritz and Bludau later noted that Eckert I and II can only be used "for sketches that need to be drafted quickly" – maybe that was the idea (after all, drawing maps was a LOT more time-consuming than it is today, so this could be).
It’s also possible that they were merely demonstrations, in the course of developing the formulae for the usable projections III to VI.

[mapnerd2022]

I wonder why he made the 1st and 2nd projections useless/novelties when from his 3rd projection onwards we have actually useful world maps...

Regrettably, don’t have the original paper in which the six Eckert projections were introduced. Zöppritz and Bludau later noted that Eckert I and II can only be used "for sketches that need to be drafted quickly" – maybe that was the idea (after all, drawing maps was a LOT more time-consuming than it is today, so this could be).
It’s also possible that they were merely demonstrations, in the course of developing the formulae for the usable projections III to VI.

Which Eckert pseudocylindrical do you prefer?

[PeteD]

Which Eckert pseudocylindrical do you prefer?

Owing to the ambiguity of the English pronoun "you", referring to both the second person singular and plural, I'm not sure whether this was intended to be addressed to Tobias in particular or intended as a general canvassing of opinion, but in case it's the latter, here's my two pennies' worth:

Short answer:

Out of the Eckert projections, I prefer Eckert IV in cases where an equal-area projection is appropriate and Eckert III otherwise since I prefer elliptical to sinusoidal or straight meridians.

Long answer:

quadibloc has recently mentioned the interrupted sinusoidal, and I agree with Milo that interrupted pseudocylindricals, including the sinusoidal, work well provided you don't mind the interruptions. I also favour pseudocylindricals in cases where the graticule is omitted since they enable you to compare the latitude at different points without one. It's sometimes stated, for example on Tobias's excellent website, that straight parallels are recommended for climatic maps since climate zones are related to latitude. Seeing as most climatic maps omit the graticule, I agree with this in most cases, but in the case of a climatic map having a graticule to show the latitude, I don't see why it couldn't have curved parallels.

However, for world maps interrupted along only one meridian and having a graticule, unless there's some other specific reason to have straight parallels for the particular usage, I'd always prefer a lenticular projection over a pseudocylindrical one since the curved parallels significantly reduce shear at the outer meridians. The compromise Eckert projections don't really bother me since they don't seem to be so widely used, but you often see Eckert IV and VI, along with the Robinson projection, used in maps having a graticule and lacking any other apparent reason to have straight parallels, where a lenticular projection would seem to be more appropriate. Note that unlike meridians, parallels other than the equator are not straight lines on the globe.

So in cases where an equal-area projection is appropriate, I'd prefer something like Frančula XIV over any Eckert projection, and in cases where the projection doesn't need to be conformal or equal-area, I'd prefer something like Canters W13 over any Eckert projection.

[Milo]

Which Eckert pseudocylindrical do you prefer?

If I had to choose, I'd agree with PeteD that III/IV are better than the others, depending on if area preservation is a goal. I/II fail smoothness at the equator, while V/VI are overly bent at the edges.

However, I dislike pseudocylindrical projections with pole lines as a rule, and don't really see the point. If you look at Eckert IV, you'll see that it's basically intermediate between Mollweide and Smyth-Craster. The difference is most pronounced in North America, which is less slanted than in the former but more than in the latter, with no real rationale for favoring that exact level of slanting.

Eckert VI looks even more similar to Mollweide, but with an uglier outer perimeter.

It's sometimes stated, for example on Tobias's excellent website, that straight parallels are recommended for climatic maps since climate zones are related to latitude. Seeing as most climatic maps omit the graticule, I agree with this in most cases, but in the case of a climatic map having a graticule to show the latitude, I don't see why it couldn't have curved parallels.

Even if you're drawing the graticule, it can still be beneficial for it to have an intuitive shape. The Dymaxion map doesn't stop being horribly confusing just because you're drawing the graticule.

I'll also note that climate has knock-on effects on numerous other things, and so is relevant to many other types of maps as well.

Lenticular projections aren't as bad, of course, but given latitude's extreme importance to basically everything, I still prefer pseudocylindrical ones unless having straight parallels is completely impractical (like a map of Antarctica, in which case you really have no choice but to use an azimuthal projection - which is fine, since concentric circles are the next simplest shape after parallel lines).

[quadibloc]

I wonder why he made the 1st and 2nd projections useless/novelties when from his 3rd projection onwards we have actually useful world maps...

My guess had always been that those projections existed to help people understand his other projections; it was easier for him to explain how the later ones worked if he had the earlier ones as more understandable examples of his principles to point to.

[mapnerd2022]

I prefer the compromise ones over the equal versions only on the fact that the equivalent ones make Africa extremely skinny, even more than in the Mollweide. I don't like Africa (and the tropical regions in general) being skinny, as long as it isn't skinny to an equal area Eckert pseudocylindrical level or a Gall-Peters level. Otherwise it doesn't bother me much.

[mapnerd2022]

Which Eckert pseudocylindrical do you prefer?

If I had to choose, I'd agree with PeteD that III/IV are better than the others, depending on if area preservation is a goal. I/II fail smoothness at the equator, while V/VI are overly bent at the edges.

However, I dislike pseudocylindrical projections with pole lines as a rule, and don't really see the point. If you look at Eckert IV, you'll see that it's basically intermediate between Mollweide and Smyth-Craster. The difference is most pronounced in North America, which is less slanted than in the former but more than in the latter, with no real rationale for favoring that exact level of slanting.

Eckert VI looks even more similar to Mollweide, but with an uglier outer perimeter.

It's sometimes stated, for example on Tobias's excellent website, that straight parallels are recommended for climatic maps since climate zones are related to latitude. Seeing as most climatic maps omit the graticule, I agree with this in most cases, but in the case of a climatic map having a graticule to show the latitude, I don't see why it couldn't have curved parallels.

Even if you're drawing the graticule, it can still be beneficial for it to have an intuitive shape. The Dymaxion map doesn't stop being horribly confusing just because you're drawing the graticule.

I'll also note that climate has knock-on effects on numerous other things, and so is relevant to many other types of maps as well.

Lenticular projections aren't as bad, of course, but given latitude's extreme importance to basically everything, I still prefer pseudocylindrical ones unless having straight parallels is completely impractical (like a map of Antarctica, in which case you really have no choice but to use an azimuthal projection - which is fine, since concentric circles are the next simplest shape after parallel lines).

Well, yes, the Earth's poles are points, not lines, that's why I was shocked when I first read that flat-polar pseudocylindricals had less angular deformation than the pointed-polar ones. It just seemed like a contradiction to me precisely because of Earth and and the globes showing the poles as points.

[Atarimaster]

My guess had always been that those projections existed to help people understand his other projections; it was easier for him to explain how the later ones worked if he had the earlier ones as more understandable examples of his principles to point to.

That’s what I meant when I said “in the course of developing the formulae”, but you put it into better words.

Which Eckert pseudocylindrical do you prefer?

Well, Eckert IV has a better distribution of distortion (in my opinion), but aesthetically speaking, I prefer Eckert VI. At least summa summarum – Eckert IV does have a better Australia. However, if I wanted to use a pseudocylindrical equal-area projection with a pole line (I usually don’t), I’d prefer Hufnagel 10 over Eckert IV and Wagner I over Eckert VI.

And generally I prefer equal-area projections over the ones with equally spaced parallels. Exceptions confirm the rule.

Edit:
Projections with equally spaced parallels lead to areal inflations that are usually too large for my taste. That’s why I like the Umbeziffern so much, it gives you the chance to improve on angular deformations, but stop at a point of areal inflations where you still feel comfortable.

[Milo]

However, if I wanted to use a pseudocylindrical equal-area projection with a pole line (I usually don’t)

Likewise, although part of the reason I like the Mollweide projection is because its "rounded pole" offers a compromise between the "angled pole" as seen in the sinusoidal projection, and the "flat pole" as seen in the Eckert or Equal Earth projections.

Cylindrical projections, of course, always have pole lines (unless they're infinite like the Mercator), but I find it more acceptable there. It helps that there's a logical reason for it, rather than just picking an arbitrary number between 0 and 1 for the relative length of the pole line to the equator.

I’d prefer Hufnagel 10 over Eckert IV

Looking at them, they seem very similar, but Hufnagel is slightly closer to cylindric/rectangular?

That just reinforces my idea that beyond a pole line of a certain length (or even sufficiently wide rounded poles, like Tobler hyperelliptical with large exponents), you might as well stop bothering with pseudocylindric projections and just accept the advantages (and disadvantages) that come with an outright cylindrical map.

And generally I prefer equal-area projections over the ones with equally spaced parallels. Exceptions confirm the rule.

Well, yeah. Having equally-spaced parallels is not by itself a particularly useful property. In the case of the plate carree or azimuthal equidistant projections, it just happens to result in other noteworthy properties (such as the equidistant property itself, as well as my resolution-efficiency metric for optimal compromise projections), but these properties do not extend to pseudocylindrical projections with equally-spaced parallels. Although it may still produce decent compromise projections at a glance, simply due to their similarity to the plate carree projection, there's no real motivation to prefer them over other comprimise projections.

And then there's the sinusoidal projection, which is equal-area and has equally-spaced parallels

(It also occurs to me that this thread is drifting rather far off from its original subject of the Eisenlohr projection. Perhaps a thread split might be a good idea, Daan?)